Bieberbach's conjecture

The Bieberbach hypothesis is a mathematical theorem in the field of complex analysis about analytical functions . It was in the year 1916 by Ludwig Bieberbach as presumption established and in 1985 by Louis de Branges de Bourcia proved and therefore since then set of de Branges called.

formulation

The Bieberbach hypothesis states that for an analytic and injective function (so-called simple function )

{\ displaystyle f \ colon \ mathbb {D} \ rightarrow \ mathbb {C}}

with ,

{\ displaystyle f (z) = z + a_ {2} z ^ {2} + a_ {3} z ^ {3} + \ ldots}

where the unit disk denotes, always ${\ displaystyle \ mathbb {D}}$

{\ displaystyle \ left | a_ {n} \ right | \ leq n}

applies to all .

{\ displaystyle n \ in \ mathbb {N}}

history

Bieberbach proved . Charles Loewner (1917) and Rolf Nevanlinna (1921) independently proved the conjecture for star-like functions. These are simple functions in the unit disk with , whose image is a star region , which is equivalent to fulfilling the Nevanlinna criterion ( has positive real parts for and ). In 1923 Loewner proved with the Loewner equation that . Later work also mostly used Loewner's method (which also played an important role in the Schramm-Löwner evolution ). In 1925 John Edensor Littlewood proved an upper bound with , Euler's number . The barrier was later improved (Milin (1965), ). Paul Garabedian and Max Schiffer handled the case (1955), Pedersen and Schiffer (1972) and Pedersen (1968) and Ozawa (1969) independently . Walter Hayman achieved asymptotic results. He showed that exists and except for a Koebe function . He also showed that for every function in the Bieberbach conjecture there are at most a finite number of exceptions. Louis de Branges finally proved the Bieberbach conjecture in 1984 about a conjecture by Isaak Moissejewitsch Milin , and the Leningrader function theory school of Milin also played the decisive role in the verification of de Brange's proof (among others Galina Wassiljewna Kusmina , Arcadii Z. Grinshpan) . The original proof of De Branges used functional analysis (but also for example the Löwner equation) and the Russian mathematicians looked for a proof without functional analysis using only methods of geometric function theory, which they then convinced De Branges to publish. ${\ displaystyle | a_ {2} | \ leq 2}$ ${\ displaystyle h (z)}$ ${\ displaystyle h (0) = 0}$ ${\ displaystyle \ textstyle {\ frac {dh} {dz}} (0) = h '(0) = 1}$ ${\ displaystyle \ textstyle k (z) = z {\ frac {h '(z)} {h (z)}}}$ ${\ displaystyle | z | <1}$ ${\ displaystyle k (0) = 1}$ ${\ displaystyle | a_ {3} | \ leq 3}$ ${\ displaystyle | a_ {n} | \ leq n \ cdot e}$ ${\ displaystyle e = 2 {,} 718 \ ldots}$ ${\ displaystyle | a_ {n} | \ leq n \ cdot 1 {,} 243}$ ${\ displaystyle n = 4}$ ${\ displaystyle n = 5}$ ${\ displaystyle n = 6}$ ${\ displaystyle \ textstyle a = \ lim _ {n \ to \ infty} | {\ frac {a_ {n}} {n}} |}$ ${\ displaystyle a <1}$

De Branges proved an inequality assumed by IM Milin in 1971 (which in turn is based on inequalities by Milin and Lebedew of 1967), from which, according to Milin, follows a conjecture by MS Robertson (1936), which in turn leads to Bieberbach's conjecture.

An alternative proof of the Milin conjecture comes from L. Weinstein

Original work

L. Bieberbach: On the coefficients of those power series that convey a simple representation of the unit circle , session. Preuss. Akad. Wiss., 1916, pp. 940-955, online in the Internet Archive
L. de Branges: A Proof of the Bieberbach Conjecture. Acta Mathematica, Vol. 154, 1985, pp. 137-152, online at projecteuclid.org
K. Löwner: Investigations into simple conformal images of the unit circle. I. , Mathematische Annalen, Volume 89, 1923, pp. 103-121, online at the Göttingen Digitization Center

literature

P. Duren, D. Drasin, A. Bernstein, A. Marden: The Bieberbach Conjecture: Proceedings of the Symposium on the Occasion of the Proof. Providence, RI: Amer. Math. Soc., 1986.
S. Gong: The Bieberbach Conjecture. Providence , RI: Amer. Math. Soc., 1999.
Christian Pommerenke The Bieberbach Conjecture , Mathematical Intelligencer Vol. 7, 1985, p. 23
OM Fomenko, GV Kuzmina : The last 100 days of the Bieberbach conjecture , Mathematical Intelligencer, Vol. 8, 1986, No. 1
J. Korevaar: Ludwig Bieberbach's conjecture and its proof by Louis de Branges , The American Mathematical Monthly, Volume 93, 1986, pp. 505-514, pdf
Paul Zorn The Bieberbach Conjecture , Mathematics Magazine, Volume 59, 1986, pp. 131-148

Web links

Bieberbach conjecture, Mathworld (English)
Wolfram Koepf: From Bieberbach's conjecture to de Branges' theorem and Weinstein's variant of proof. (PDF; 205 kB)

Individual evidence

↑ Löwner, Investigations on the distortion in conformal mappings of the unit circle , which are supplied by functions with non-vanishing derivation ${\ displaystyle | z | <1}$ , session reports Gesellschaft der Wiss. Leipzig, Volume 69, 1917, pp. 89-106
^ Nevanlinna, On the conformal mapping of star regions , Översikt av Finska Vetenskps-Soc. Förh., Volume 63 (A), No. 6, 1920/21, pp. 1-21
↑ Littlewood, On inequalities in the theory of functions , Proc. London Math. Soc., Vol. 23, 1925, pp. 481-519
↑ Milin, Estimation of coefficients of univalent functions , Soviet Math. Dokl., Volume 6, 1965, pp. 196-198
^ R. Garabedian, M. Schiffer, A Proof of the Bieberbach Conjecture for the Fourth Coefficient , J. Rational Mech. Anal., Volume 4, 1955, pp. 427-465
↑ Pedersen, Schiffer, A Proof of the Bieberbach Conjecture for the Fifth Coefficient , Arch. Rational Mech. Anal., Volume 45, 1972, pp. 161-193
↑ Pedersen, A Proof of the Bieberbach Conjecture for the Sixth Coefficient , Arch. Rational Mech. Anal., Volume 31, 1968/69, pp. 331-351
↑ Ozawa, On the Bieberbach Conjecture for the Sixth Coefficient , Kodai Math. Sem. Rep., Volume 21, 1969, pp. 97-128
^ WK Hayman, FM Stewart: Real Inequalities with Applications to Function Theory. Proc. Cambridge Phil. Soc., Vol. 50, 1954, pp. 250-260
↑ Milin, Univalent functions and orthonormal systems , American Mathematical Society 1977, Russian original Moscow: Nauka 1971.
^ Robertson, On the theory of univalent functions , Annals of Mathematics, Volume 37, 1936, pp. 374-408
^ L. Weinstein, The Bieberbach Conjecture , Internat. Math. Res. Notes, Vol. 5, 1991, pp. 61-64

[1] Löwner, Investigations on the distortion in conformal mappings of the unit circle , which are supplied by functions with non-vanishing derivation ${\ displaystyle | z | <1}$ , session reports Gesellschaft der Wiss. Leipzig, Volume 69, 1917, pp. 89-106

[2] Nevanlinna, On the conformal mapping of star regions , Översikt av Finska Vetenskps-Soc. Förh., Volume 63 (A), No. 6, 1920/21, pp. 1-21

[3] Littlewood, On inequalities in the theory of functions , Proc. London Math. Soc., Vol. 23, 1925, pp. 481-519

[4] Milin, Estimation of coefficients of univalent functions , Soviet Math. Dokl., Volume 6, 1965, pp. 196-198

[5] R. Garabedian, M. Schiffer, A Proof of the Bieberbach Conjecture for the Fourth Coefficient , J. Rational Mech. Anal., Volume 4, 1955, pp. 427-465

[6] Pedersen, Schiffer, A Proof of the Bieberbach Conjecture for the Fifth Coefficient , Arch. Rational Mech. Anal., Volume 45, 1972, pp. 161-193

[7] Pedersen, A Proof of the Bieberbach Conjecture for the Sixth Coefficient , Arch. Rational Mech. Anal., Volume 31, 1968/69, pp. 331-351

[8] Ozawa, On the Bieberbach Conjecture for the Sixth Coefficient , Kodai Math. Sem. Rep., Volume 21, 1969, pp. 97-128

[9] WK Hayman, FM Stewart: Real Inequalities with Applications to Function Theory. Proc. Cambridge Phil. Soc., Vol. 50, 1954, pp. 250-260

[10] Milin, Univalent functions and orthonormal systems , American Mathematical Society 1977, Russian original Moscow: Nauka 1971.

[11] Robertson, On the theory of univalent functions , Annals of Mathematics, Volume 37, 1936, pp. 374-408

[12] L. Weinstein, The Bieberbach Conjecture , Internat. Math. Res. Notes, Vol. 5, 1991, pp. 61-64